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%I #27 Jun 09 2019 19:24:59
%S 1,1,0,1,1,0,1,3,3,0,1,6,17,17,0,1,10,55,155,155,0,1,15,135,736,2073,
%T 2073,0,1,21,280,2492,13573,38227,38227,0,1,28,518,6818,60605,330058,
%U 929569,929569,0,1,36,882,16086,211419,1879038,10233219,28820619
%N Triangular array A065547 unsigned and transposed.
%D Louis Comtet, Analyse Combinatoire, PUF, 1970, Tome 2, pp. 98-99.
%H J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
%H J. M. Hammersley, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/14_1_1.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23.
%F Sum_{k >= 0} (-1/2)^k*T(n, k) = (1/2)^n.
%F Sum_{k >= 0} (-1/6)^k*T(n, k) = (4^(n+1)- 1)/3*(6^n).
%F Equals A000035 DELTA [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ...], where DELTA is Deléham's operator defined in A084938.
%F T(n,n-1) = A110501(n), Genocchi numbers of first kind of even index. - _Philippe Deléham_, Feb 16 2007
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 3, 3, 0;
%e 1, 6, 17, 17, 0;
%e 1, 10, 55, 155, 155, 0;
%e ...
%t h[n_, x_] := Sum[c[k]*x^k, {k, 0, n}]; eq[n_] := SolveAlways[h[n, x*(x-1)] == EulerE[2*n, x], x]; row[n_] := Table[c[k], {k, 0, n}] /. eq[n] // First // Abs // Reverse; Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Oct 02 2013 *)
%Y Row sums sum(k>=0, T(n, k)) = A006846(n), values of Hammersley's polynomial p_n(1).
%Y Sum(k>=0, 2^k*T(n, k)) = A005647(n), Salie numbers.
%Y Sum(k>=0, 3^k*T(n, k) = A094408.
%Y Sum(k>=0, 4^k*T(n, k) = A000364(n), Euler numbers.
%Y Cf. A006846, A005647, A000364, A065547, A000795.
%Y Cf. A006846 A005647 A000364 A065547 A000795 A084938.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Jul 19 2003
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